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 The Triangle's Bottom (Posted on 2008-02-05)
Let T be the set of triangular numbers and T* be the set of all products of any two triangular numbers. Show that:

1. Among elements of T, each of the digits 0,1,5 and 6 occur in the units place twice as frequently as each of the digits 3 and 8. (More precisely, if MBk is the set of elements of T that are less than B and end in k, then, e.g., MB1/MB8 approaches 2 as B approaches infinity.)

2. None of the elements of T* end in 2 or 7.

 See The Solution Submitted by FrankM Rating: 3.0000 (1 votes)

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 solution | Comment 1 of 4

Each triangular number T(i) equals the previous triangular number plus i:  T(i) = T(i-1)+i.

So:

` i   T(i) 1     1 2     3 3     6 4    10 5    15 6    21 7    28 8    36 9    4510    5511    6612    7813    9114   10515   12016   13617   15318   17119   19020   21021   23122   25323   27624   30025   32526   35127   37828   40629   43530   46531   49632   52833   56134   59535   63036   66637   70338   74139   78040   820`

Note that there is a cycle of 20 in the last digits. For i itself, there's a cycle of 10 in the last digits, but it is not until i=21 that T(i) also returns to a last digit of 1, as at T(1). From then on, due to the nature of addition, the length-20 cycle repeats.

In the cycle of 20, there are the following counts of last digits:

`finaldigit  count 0      4 1      4 2      0 3      2 4      0 5      4 6      4 7      0 8      2 9      0 `

Therefore, over the long run, the occurrence of each of 0, 1, 5 and 6 is twice as frequent as either of 3 or 8.

When taking the products of these numbers, which never end in 2, 4, 7 or 9, if either one ends in a zero, the product will end in a zero. If either ends in a 1, the product will end in the same digit as the other triangular number and never be 2, 4, 7 or 9.  The other possible pairs of ending digits result in products ending as follows:

`1  3  5  6  83  9  5  8  45  5  5  0  06  8  0  6  88  4  0  8  4`

None of these is a 2 or a 7.

 Posted by Charlie on 2008-02-05 14:48:22

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